Wrapping a path around three cylinders
What you can do is temporary extruding the path vertices (as it is a mesh if I see well), then apply the shrinkwrap modifier and delete the previously created vertices.

Make so just add surfaces so that the modifier can do a projection (mode 'project/on surface' and 'back'):



Projection wll be done along the normals, so I think here you need to extrude along Z. If not the path won't stay horizontal.



Edit (if that matters...): I previously misunderstood the question, thinking you wanted to tangent the path around the cylinders. This is not the case, but, eventually, this approach can interest someone:

Transform the cylinders into a convex hull: